Unbiased estimate

Fig. 2 Simulated example of DR/LS detection (after Ref. 22). A) Tme and measured mass chromatograms [w (V), w(V)] three samples of the BB function g(V, V) and estimated corrected chromatogram [ (V)]. B) True and measured molar mass chromatograms [. s (V), rLs(V)] and estimated corrected molar mass chromatogram fLs (F)-C) Unbiased linear calibration [log A/(V)1 estimated ad hoc calibration [log M (V)] estimated unbiased calibration [log A/(V)], and estimated linear unbiased calibration [log AfiinXV)]. D) True MMD [w (log A/)] MMD estimate obtained from >v(V) and log M V) [w(log Af)] and MMD estimate obtained from and log Afun.(V)...

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of O, not s. Although is an unbiased estimator of O, the value of for any randomly selected sample may differ significantly from O. To account for the uncertainty in estimating O, the term z in equation 4.11 is replaced with the variable f, where f is defined such that f > z at all confidence levels. Thus, equation 4.11 becomes... 

ROBUST AND UNBIASED ESTIMATIONS IN CHEMICAL DATA TREATMENT... 

If the sample is unbiased, estimate the source mean, so that... 

Finally we need to compare the variance of our estimator with the best attainable. It can be shown that The Cramer-Rao lower bound (CRLB) is a lower bound on the variance of an unbiased estimator (Kay, 1993). The quantities estimated can be fixed parameters with unknown values, random variables or a signal and essentially we are finding the best estimate we can possibly make. 

Note that by unbiased we mean that when the noise tends to zero we expect a —> a. The left-hand side can be written simply as Var(d), the variance of the estimate, since the estimator is unbiased. The CRLB for some PDFs of interest is now computed. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

The least squares estimator has several desirable properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

The formula for calculating the budget of a GP practice must offer an unbiased estimator of the expected level of expenditure if each GP practice had a standard response to the needs of its population. Even if the considerable technical difficulties of establishing this formula could be overcome, the actual expenditure of a GP practice would differ from the budgeted amount due to characteristics of the patients not taken into account in the formula (socioeconomic characteristics, chronic diseases, private coverage and so on), variations in clinical practice between GP practices, random variations in the level of disease and price variations. For a population of 10 000 inhabitants (a reasonable mode for a GP practice) there is a one-third probability that the actual expenditure will deviate more than 10 per cent from a well-designed budget.22... 

Instead of estimating -/ 1 ln(exp(-/W(f)) directly using (5.44), one can use cumulant expansion approaches, as in regular free energy perturbation theory (see e.g., [20, 39] for combining cumulant expansions about the initial and final states). Unbiased estimators for cumulants can be used. Probably the most useful relations involve averages and variances of the work ... 

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

Finally, it is interesting to note that biases can be introduced by data fitting at low counts even with the use of ordinarily unbiased estimators like the maximum likelihood estimator [37],... 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

The arithmetic mean Z has excellent estimation performance. The estimation Z is unbiased, which means... 

These CFAR procedures suffer from the fact that they are specifically tailored to the assumption of uniform and homogeneous clutter inside the reference window. Based on these assumptions, they estimate the unknown clutter power level using the unbiased and most efficient arithmetic mean estimator. Improved CFAR procedures should be robust with respect to different clutter background and target situations. Also in non-homogeneous situations CFAR techniques should remain able to provide reliable clutter power estimations. 

If it is assumed that the measurement errors are normally distributed, the resolution of problem (5.3) gives maximum likelihood estimates of process variables, so they are minimum variance and unbiased estimators. 

The Unbiased Estimation Technique (UBET) was developed by Rollins and Davis (1992). This approach simultaneously provides unbiased estimates and confidence intervals of process variables when biased measurements and process leaks exist. 

Rollins, D., and Davis, J. (1992). Unbiased estimation of gross errors in process measurements. AlChE J. 38,563-572. 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

In robust statistics, rather than assuming an ideal distribution, an estimator is constructed that will give unbiased results in the presence of this ideal distribution, but that will try to minimize the sensitivity to deviations from ideality. Several approaches are described here ... 

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